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Complex dynamics in Quantum Dot Light Emitting Diodes
K. A. Al Naimee
S.F. Abdalah
R. Meucci
Istituto Nazionale di Ottica- CNR,
Largo E. Fermi 6, 50125, Florence,
Italy
Istituto Nazionale di Ottica- CNR,
Largo E. Fermi 6, 50125, Florence,
Italy
Istituto Nazionale di Ottica- CNR,
Largo E. Fermi 6, 50125, Florence,
Italy
Dept. of Physics, College of
Science, University of Baghdad, Al
JAdiriyah, Baghdad, Iraq
A. Al Khursan
H. Al Husseini
Dept. of Physics, College of
Science, University of Baghdad, Al
JAdiriyah, Baghdad, Iraq
Dept. of Physics, College of
Science, Thi-Qar University,
ThiQar, Iraq
A. H. Khedir
F.T. Arecchi
Istituto Nazionale di Ottica- CNR,
Largo E. Fermi 6, 50125, Florence,
Italy
Dept. of Physics, College of
Science, University of Baghdad, Al
JAdiriyah, Baghdad, Iraq
Abstract
We report an experimental and theoretical investigation on the
appearance of Mixed Mode Oscillations (MMOs) and chaotic
spiking in a Quantum Dot Light Emitting Diode (QDLED). The
underlying dynamics is completely determined by the variation
of the injecting bias current in
the wetting layer of the
QDLED. The theoretical model qualitatively reproduces the
experimental results.
Keywords: Quantum Dot LED, Mixed Mode Oscillations,
Chaos, Control
1 Introduction
Most light sources exhibit intensity and phase
fluctuations due to the nature of the quantum transition
process itself. These fluctuations are very important when
the stability is a priority. In fact, every spontaneous
emission event in the oscillating mode causes a phase
variation of the electromagnetic field. This variation is
defined as a quantum noise and it is responsible forcarrier
density variations.
Semiconductor nitrides such as aluminum-nitride (AlN),
gallium-nitride (GaN), and indium-nitride (InN) are
commonly utilized as materials for their potential use in
high-power and high-temperature optoelectronic devices.
The commonly quoted value for the optical band gap of
InN was 1.89 eV[Al-Husseini H, Al-Khursan AH, AlDabagh, 2009]. Nevertheless other values are obtained by
changing the state [Chuang, 1996].The properties of IIIN QDs are closely related to those of bulk materials [
Steiner, 2004]. Although bulk LEDs can be found as
wurtzite, zinc blend, and rock salt structures, the wurtzite
structure is more dominant in thermodynamically stable
condition. III-nitride QDs are commonly strained
systems. [Jindal, 2011]
In addition to chaotic spiking, recent studies involving
surface chemical reactions[ Bertram and Mikhailov,
2001:Bertram,
2003:Kim,2001],
electrochemical
systems[Kope, 1995: Plenge, Rodin, Scholl, andKrischer
, 2001]neural and cardiac cells[ Alonso andLlin´as, 1989:
Medvedev andCisternas, 2004], calcium dynamics[U.
Kummer, 2000] and plasma physics[Mikikian,
Cavarroc,Couedel, Tessier, and Boufendi, 2008] showed
thatoscillatory dynamics often takes place in the form of
complex temporal sequences known as Mixed Mode
Oscillations (MMOs)[Marino, Ciszak, Abdalah, AlNaimee, Meucci, and Arecchi, 2011]. However, periodicchaotic sequences and Farley sequences of MMOs do not
necessarily involve a torus or a homoclinicorbit. They
can occur also through the canard phenomenon [Brøns,
Krupa, andWechselberger, 2006]. Here a limit cycle born
at a supercritical Hopf bifurcation experiences the abrupt
transition from a small-amplitude quasi-harmonic cycle
to large relaxation oscillations in a narrow parameter
range. Most studies of this dynamics have been carried
out in chemical systems. Nevertheless in semiconductor
laser systems with optoelectronic feedback incomplete
homoclinic scenarios have been recently predicted and
observed [Al-Naimee, Marino, Ciszak, Meucci, and
Arecchi, 2009: Al-Naimee et al.,2010].
The main goal of this work is to provide a physical model
reproducing qualitatively the experimental results and
showing that chaotic spiking and MMOs are a
consequence bias current variations. Our experiments
reveal that spiking competition in the active layer
enhances the influence of self-feedback and modulating
perturbation thus opening up new avenues for the study
of chaos in quantum systems.
2 Experiment
Spikingphenomena in the output of the QDLEDs are
related to a feedbackprocess that couples the output
signal partially back to the input [Albert, 2011].We
consider an open-loop optical system, consisting of a
QDLED source emitting at 650nm wavelength. The
QDLED operates with variable driven current. The light
is sent to a photodetector whose output current is
proportional to the optical intensity. The corresponding
signal is sent to a variable gain amplifier characterized by
a nonlinear transfer function of the formf(w) = Aw/(1 +
𝛽𝑤), where A is the amplifier gain and 𝛽a saturation
coefficient, then to a fast (500 MHz) digital oscilloscope.
By decreasing the dc-pumping current which plays the
role of control parameter, we observe the dynamical
sequence shown in figures 1(a)-(f). In figure (1-a) we
show the regular spiking at the injection current of 6.2
mA. Changing the control parameter, a transition from
periodic dynamics toMMOs is observed as shown in
figure 1-(b-d). When the injection current reaches a value
of 5.01mA, the detected optical output is in a chaotic
spiking regime where large intensity pulses are separated
by irregular time intervals in which the system displays
small-amplitude chaotic oscillations[see figure1-e].
Typical time traces are characterized by a mixture of
large-amplitude relaxation spikes (L) followed by smallamplitude (S) quasi-harmonic oscillations, where
oscillations intermediate between L and S do not occur).
As the injection current is farther decreased (4.7mA), the
spiking behavior vanishes and eventually a steady state is
established as shown in figure 1-f. This scenario is
qualitatively similar to Homoclinic Chaos (HC) as
previously observed in semiconductor lasers and LEDs
with optoelectronic feedback [Al Naimee, Marino,
Ciszak, Meucci, andArecchi, 2009]. In HC signals, the
pulse duration (associated with a precise orbit in the
phase space) is uniform, while the interpulse times vary
irregularly. This is shown by the corresponding
interspike interval (ISI) probability distribution (fig.1 e).
The ISI histogram displays sharp peaks,superimposed to
a continuous background, corresponding to unstable
periodic orbits embedded in the chaotic attractor [Alonso
andLlin´as, 1989]. We will show in the next section that
a fully deterministic theoretical model explains such a
feature.
I
P(ISI)
n
t.
(
I
t
I Figure 1 Transition from periodic toSchaotic spiking to
)
steady state as the dc-pumping current
is varied (a)
𝐧𝐭.and
(𝒕 −
I
6.2
mA,
(b)
5.52
mA,
(c)
5.35mA,
(d)
5.21
mA, (e)
(
𝟐𝝉)
5.01mA, (f) 4.700 mA. (g). Experimental
reconstruction
s
of the phase portrait through the embedding
technique.
)
(h) The corresponding experimental ISI probability
distribution for the chaotic spiking regime.
3 The model
In this study, we propose a rate equations model for a
QDLED considered as a three-level system(see figure 2).
In the QDLED, the electrons are first injected into the
wetting layer (WL) before they are captured by the QDs.
We consider a system made up of upper and lower
electronic levels.
The equations describing the dynamics of the total
populationnQD of carriers in the upper levels, the number
of photons in the optical modes, and wetting layer
population nwl are as follows
𝑛̇ 𝑄𝐷 = γc nwl (1 −
ṅ wl
2Nd
In this section we investigatethe unstable behavior from
the QDLED equations without external feedback. To do
so, we rescale the system (Eq.s1) to a set of
dimensionless equations. Defining new variables and
A
𝑠̇ = AnQD − ds − Γs
nQD
whereno is the occupation number of the lower level. In
such a case, the spontaneous emission coefficient and
absorption coefficient possess identical line shapes. For
realistic material systems, such as semiconductor or
organic emitters, both the lower and the upper levels can
be considered as inhomogenously broadened. Population
distributions in the lower and upper levels have to be
taken into account explicitly in order to determine the
correct relation between absorption and spontaneous
emission spectra [Casey and Panish, 1978].
dimensionless parameters asx = s, y = (n − no s),w =
) − γr nQD − (AnQD − −ds )(1)
nwl γc
A
γc
nQD
J
= − γn nwl − γc nwl (1 −
)
e
2Nd
γn
, γn =
t
𝑡`
γ=
Γ
γn
, γ1 =
A
A
γn
, Nd ≡ a, no ≡ b and δo =
Γ
, γ2 = , γ3 =
Γ
J
Ae
,
γr
γn
, γ4 =
Eqs. 1 can be
rewritten as
ẋ = γ(y − x)
E
ne
rg
y
ẏ = γ1 γ2 w(1 − bx⁄a) − γ1 y(1 + w⁄a) − bγ1 (x − y) − γ3 (3)
WL
ẇ = γ4 δo − w(1 −
QD
yγ4
⁄aγ2 ) − γ4 w(1 − bx⁄a)
Here prime means differentiation with respect to t`and
the bias current density is represented byδo .
In our simulations, the injection current in the pumping
WL
Z
Figure 2 Energy diagram illustrates the two
recombination mechanisms considered in this work of the
active layer QDLED; recombination radiative and nonradiative via deep level and reabsorption recombination
processes
where A is the spontaneous emission rate into the optical
mode, γr and γn are the non-radiative decay rates of the
population of carriers in the upper levels and WL
respectively, Nd is the two-dimensional density of dots;
J is the injection current, e is electron charge, γc is the
capture rate from WL into an empty dot, and d and Γ are
the absorption and output coupling rate of photons in the
optical mode, respectively.
For a two-level atomic system where the transition is
homogeneously broadened, it can be shown [Loudon,
1983] that
d = Ano (2)
J
term , has been considered as the sum a dc component
e
and an ac component in order to consider an additional
periodic modulation
J = Io + Iac sin(2πfm t)
(4)
4 Numerical results
Equations 3 are solved numerically using the fourth order Runge-Kutta method with 0.01 ns as step time. The
parameter values used in the simulations are (the initial
conditions are assigned as xo = 0.066, yo = 0.99, wo =
0.0049), γ =0.127, γ1 = 0.144, γ2 = 0.03, γ3 = 0.07, γ4
=0.087, a = 1.04 and b= 3.838. The injection parameters,
i.e., the dc bias current,Io , or the dc bias strength , the
modulation current , Iac , or modulation depth, and
modulation frequency, f m , have been chosen carefully.
As in the experiment, the chaotic spiking regime arises
from the interplay of the large phase-space orbits and a
period doubling route to chaos. In figure3 we show
several time series and the corresponding attractorsfor
different current injection current δo values. The related
ISI distributions are reported in figure 4.
wo = 0.0049, γ = 0.127, γ1 = 0.144, γ2 = 0.03, γ3 =
0.07, γ4 = 0.087, a = 1.04 and b = 3.838
The WL has a large number of states it can be considered
as a continuum state and its carrier number is always
higher than zero as. There are limited number of states in
QD structure, then the ground state is filled and emptied
due to the capturing and relaxation processes (positive
 c and negative  r parts in the second equation of the
system, Eqs. 1). Thus, the behavior of the carrier
numbernqd in QD goes between positive and negative
values as shown in the left panel of figure 3. So, the
negative part of photon number shown in figure 3 is due
to both absorption and emission and it is also shown in
experimental works in bulk LED.
In Figure 3 we observe that the transition from periodic
dynamics to chaos is accompanied by MMOs. In these
regimes, a mixture of L large amplitude relaxation spikes
followed by S small-amplitude quasi-harmonic
oscillations.
X
0
-0.2
W(t)
(
a
)
0.2
Figure 4 The ISI distribution of the conditions as in fig. 3
,when a) δo=6, b) δo=4, c) δo=0.5, d) δo=0.1.
5
0
0.5
0
-0.5
-1
X(t)
-0.4
500
1500
(
b
)
0
-0.2
-0.4
10
W(t)
0.2
X
1000
Time
5
0
1
0
-0.6
X(t)
-0.8
500
1500
-1
-1
-1
20
0
2
-2
0
500
1000
Time
-0.5
-1
-2
-2
-4 -3
X(t)
Y(t)
1500
(
d
)
10
Figure 5The Bifurcation diagram for the model
equations. The parameter values are set as in figure 3.
0
1
-1
0
500
1
0
20
W(t)
0
X
0
-0.5
Y(t)
In Figure 5 we show the bifurcation diagram of the
maxima of x variable as a function of δo starting from
the Hopf bifurcation. As δo approaches values less than
1, chaotic amplitude fluctuations are sufficiently large to
trigger fast spiking dynamics. This results in an erraticsensitive
to
initial
condition-sequence
of
homoclinicspikeson top of a chaotic background.
40
W(t)
X
1000
Time
(
c
)
0
0.2
0
-0.4 -0.2
Y(t)
1000
Time
1500
-0.5
-1
-1
-2 -1.5 Y(t)
X(t)
0
Figure 3 Numerical simulations of the model equations.
Left panels: time series for the light intensity. Right
panels: phase space reconstruction of the attractors.(a)=6,
δ0 (b) =4, δ0 (c) =0.5, δ0 (d) = 0.1 correspond to δ0 = 60.1. The other parameters are: xo = 0.066, yo = 0.99,
As introduced, we have also considered the spiking
dynamics in presence of a modulation amplidudeIac. As
in the experiment, increasing the modulation depth by
around 2 leads to faster chaotic spiking regimes until the
duration of the slow–fast pulses becomes of the order of
the chaotic background characteristic timescale (figure
6).
Medvedev G. S. and Cisternas J. E (2004)Physica D 194,
333.
3
Kummer U. et al (2000)Biophys. J. 79, 1188.
2
Mikikian
M.,
Cavarroc
M.,
Couedel
L.,
TessierY.andBoufendi L.(2008) Phys. Rev. Lett.
100,225005.
X
m
2.5
1.5
1
0.5
0
Alonso A.A. and Llin´as R.(1989) Nature (London) 342,
175.
0.5
1
1.5
2
2.5
Iac
Figure 6 Bifurcation diagram of the photon density as a
function of modulation depth (Iac) for the dc bias strength
value δo= 0.6 and modulation frequency f m  2.4 102 .
5 Conclusions
In conclusions, we have numerically and experimentally
demonstrated the existence of chaotic behavior and
MMOs in the dynamics of a quantum dot light emitting
diode with optoelectronic feedback.
We provide a feasible minimal model reproducing
qualitatively the experimental results showing chaotic
and mixed mode oscillations dynamics. Remarkably,
chaotic dynamics can be controlled by changing the bias
current with a suitable modulation.
.
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